My astronomy teachers never would answer this for me... Redshift obviously indicates an object (such as another galaxy) is moving away, but how do we know its acceleration from this? It's my understanding that the instantaneous redshift of an object is an indicator solely of its instantaneous velocity, and we have certainly not been observing redshift long enough to compare and determine a change in velocity

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You're right, a redshift will give you a measure of a velocity, not an acceleration. I'm assuming when you say acceleration, you're referring to the accelerating expansion of the universe so you're not looking at the particular acceleration of an object (which you could get from examining the force of gravity imparted on that object).

So in addition to a redshift, you need to find a way to independently measure the distance. To get out to an appreciable redshift (z~0.5), Type Ia supernova are used as "standard candles". They are considered standard because they're believed to be from white dwarfs that explode because they have reached their maximum possible mass, the Chandrasekhar limit, of 1.4 solar masses. Therefore, when they explode, they should all look about the same and any difference in brightness would be from how far they are away from us. So they measured the distance and the redshift of the galaxy the supernova originated from to determine the expansion of the universe appears to be accelerating over time.

But couldn't thousands of other factors be tinting and filtering the photons before they reach us? We know from sunsets and sunrises that red light (or lower wavelengths in general) last longer and travel farther than the rest. Couldn't the galaxies just appear red because they're so insanely far away, and only red light made it this far?
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SupuhstarJul 7 '14 at 22:19

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Red light doesn't "last longer" or "travel father" than blue light. Sunrise/sunset and blue skies are caused by Rayleigh scattering, which is a particular process that preferentially scatters blue light. Also, when redshifts are measured precisely, they are measured with sets spectral lines which have a known wavelenth so you can see how far the lines have shifted. The blue light doesn't just go away, it actually appears in the red. This is the same doppler effect for when the pitch of passing vehicles changes coming towards vs away from you.
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AaronJul 8 '14 at 21:11

I don't know the exact details (so I'll be leaving the math out anyway), but this is my understanding and I hope it helps you get some idea about what's being done.

When you look at galaxies farther away, you're looking at them farther back in time (because speed of light is constant). So, to sample the velocities of galaxies over time, you don't need to observe them for a long time, but you can work with galaxies that are at different distances, and hence at different times in the history of the universe.

Now if you notice that the recession velocities of galaxies that are farther away are slower than what you would expect from a constant expansion scenario (Hubble's law, for example), that means the universe was expanding slower, back then, than it is, now. If there is a clear trend that this is happening, then you can say that the speed of expansion of the universe has been increasing over time. This means that the universe is accelerating.

Of course, this is a lot more complicated, since if the universe is accelerating, Hubble's law itself becomes shaky, if not invalid (as it assumes a direct proportionality between distance and recession velocity). Also, if you cannot find distance using Hubble's law, then you have to have an independent distance measurement that far away, which is also problematic, since afaik, there aren't such distance indicators that are valid that far away (since physics is different due to changes in stellar metallicities etc.; a lot of problems start showing up). So I don't know how this is done in practice, but in theory what I described above is how one would arrive at the conclusion of an expanding universe. I would like to know how this is really done in practice, as well.